Tensor scale (t-scale) is a parametric representation of local structure morphology that simultaneously describes its orientation, shape and isotropic scale. At any image location, t-scale is the parametric representation of the largest ellipse (an ellipsoid in 3D) centered at that location and contained in the same homogeneous region. Recently, we have improved the t-scale computation algorithm by: (1) optimizing digital representations for LoG and DoG kernels for edge detection and (2) ellipse fitting by using minimization of both algebraic and geometric distance errors. Also, t-scale has been applied to computing the deformation vector field with applications to medical image registration. Currently, the method is implemented in two-dimension (2D) and the deformation vector field is directly computed from t-scale-derived normal vectors at matching locations in two images to be registered. Also, the method has been used to develop a simple algorithm for computing 2D warping from one shape onto another. Meanwhile, t-scale has been applied to generating interpolation lines with applications to medical image interpolation using normal vector. Normal vector yields local structure orientation pointing to the closest edge. However, this information is less reliable along the medial axis of a shape as it may be associated with either of the two opposite edges of the local shape. This problem is overcome using a shape-linearity measure estimating relative changes in scale along the orthogonal direction. Preliminary results demonstrate the method's potential in estimating deformation between two images and interpolating between neighboring slices in a grey scale image.

Scale is a widely used notion in medical image analysis that evolved in the form of scale-space theory where the key idea is to represent and analyze an image at various resolutions. Recently, a notion of local morphometric scale referred to as "tensor scale" was introduced using an ellipsoidal model that yields a unified representation of structure size, orientation and anisotropy. In the previous work, tensor scale was described using a 2-D algorithmic approach and a precise analytic definition was missing. Also, with previous framework, 3-D application is not practical due to computational complexity. The overall aim of the Ph.D. research is to establish an analytic definition of tensor scale in n-dimensional (n-D) images, to develop an efficient computational solution for 2- and 3-D images and to investigate its role in various medical imaging applications including image interpolation, filtering, and segmentation. Firstly, an analytic definition of tensor scale for n-D images consisting of objects formed by pseudo-Riemannian partitioning manifolds has been formulated. Tensor scale captures contextual structural information which is useful in local structure-adaptive anisotropic parameter control and local structure description for object/image matching. Therefore, it is helpful in a wide range of medical imaging algorithms and applications. Secondly, an efficient computational solution of tensor scale for 2- and 3-D images has been developed. The algorithm has combined Euclidean distance transform and several novel differential geometric approaches. The accuracy of the algorithm has been verified on both geometric phantoms and real images compared to the theoretical results generated using brute-force method. Also, a matrix representation has been derived facilitating several operations including tensor field smoothing to capture larger contextual knowledge. Thirdly, an inter-slice interpolation algorithm using 2-D tensor scale information of adjacent slices has been developed to determine the interpolation line at each image location in a gray level image. Experimental results have established the superiority of the tensor scale based interpolation method as compared to existing interpolation algorithms. Fourthly, an anisotropic diffusion filtering algorithm based on tensor scale has been developed. The method made use of tensor scale to design the conductance function for diffusion process so that along structure diffusion is encouraged and boundary sharpness is preserved. The performance has been tested on phantoms and medical images at various noise levels and the results were quantitatively compared with conventional gradient and structure tensor based algorithms. The experimental results formed are quite encouraging. Also, a tensor scale based n-linear interpolation method has been developed where the weights of neighbors were locally tuned based on local structure size and orientation. The method has been applied on several phantom and real images and the performance has been evaluated in comparison with standard linear interpolation and windowed Sinc interpolation methods. Experimental results have shown that the method helps to generate more precise structure boundaries without causing ringing artifacts. Finally, a new anisotropic constrained region growing method locally controlled by tensor scale has been developed for vessel segmentation that encourages axial region growing while arresting cross-structure leaking. The method has been successfully applied on several non-contrast pulmonary CT images. The accuracy of the new method has been evaluated using manually selection and the results found are very promising.

Neste trabalho são apresentados dois novos descritores de forma para tarefas de recuperação de imagens por conteúdo (CBIR) e análise de formas, que são construídos sobre uma extensão do conceito de tensor scale baseada na Transformada de Distância Euclidiana (EDT). Primeiro, o algoritmo de tensor scale é utilizado para extrair informações da forma sobre suas estruturas locais (espessura, orientação e anisotropia) representadas pela maior elipse contida em uma região homogênea centrada em cada pixel da imagem. Nos novos descritores, o limite do intervalo das orientações das elipses do modelo de tensor scale é estendido de 180º para 360º, de forma a melhor discriminar a descrição das estruturas locais. Então, com base em diferentes abordagens de amostragem, visando resumir informações mais relevantes, os novos descritores são construídos. No primeiro descritor proposto, Tensor Scale Sector (TSS), a distribuição das orientações relativas das estruturas locais em setores circulares é utilizada para compor um vetor de características de tamanho fixo, para uma caracterização de formas baseada em região. No segundo descritor, o Tensor Scale Band (TSB), foram considerados histogramas das orientações relativas extraídos de bandas concêntricas, formando também um vetor de características de tamanho fixo, com uma função de distância de tempo linear. Resultados experimentais com diferentes bases de formas (MPEG-7 e MNIST) são apresentados para ilustrar e validar os métodos. TSS demonstra resultados comparáveis aos métodos estado da arte, que geralmente dependem de algoritmos custosos de otimização de correspondências. Já o TSB, com sua função de distância em tempo linear, se demonstra como uma solução adequada para grandes coleções de formas. / In this work, two new shape descriptors are proposed for tasks in Content-Based Image Retrieval (CBIR) and Shape Analysis tasks, which are built upon an extended tensor scale based on the Euclidean Distance Transform (EDT). First, the tensor scale algorithm is applied to extract shape attributes from its local structures (thickness, orientation, and anisotropy) as represented by the largest ellipse within a homogeneous region centered at each image pixel. In the new descriptors, the upper limit of the interval of local orientation of tensor scale ellipses is extended from 180º to 360º, to better discriminate the description of local structures. Then, the new descriptors are built based on different sampling approaches, aiming to summarize the most relevant features. In the first proposed descriptor, Tensor Scale Sector descriptor (TSS), the local distributions of relative orientations within circular sectors are used to compose a fixed-length feature vector, for a region-based shape characterization. For the second method, the Tensor Scale Band (TSB) descriptor, histograms of relative orientations are considered for each circular concentric band, to also compose a fixed-length feature vector, with linear time distance function for matching. Experimental results for different shape datasets (MPEG-7 and MNIST) are presented to illustrate and validate the methods. TSS can achieve high retrieval values comparable to state-of-the-art methods, which usually rely on time-consuming correspondence optimization algorithms, but uses a simpler and faster distance function, while the even faster linear complexity of TSB leads to a suitable solution for very large shape collections.

Neste trabalho são apresentados dois novos descritores de forma para tarefas de recuperação de imagens por conteúdo (CBIR) e análise de formas, que são construídos sobre uma extensão do conceito de tensor scale baseada na Transformada de Distância Euclidiana (EDT). Primeiro, o algoritmo de tensor scale é utilizado para extrair informações da forma sobre suas estruturas locais (espessura, orientação e anisotropia) representadas pela maior elipse contida em uma região homogênea centrada em cada pixel da imagem. Nos novos descritores, o limite do intervalo das orientações das elipses do modelo de tensor scale é estendido de 180º para 360º, de forma a melhor discriminar a descrição das estruturas locais. Então, com base em diferentes abordagens de amostragem, visando resumir informações mais relevantes, os novos descritores são construídos. No primeiro descritor proposto, Tensor Scale Sector (TSS), a distribuição das orientações relativas das estruturas locais em setores circulares é utilizada para compor um vetor de características de tamanho fixo, para uma caracterização de formas baseada em região. No segundo descritor, o Tensor Scale Band (TSB), foram considerados histogramas das orientações relativas extraídos de bandas concêntricas, formando também um vetor de características de tamanho fixo, com uma função de distância de tempo linear. Resultados experimentais com diferentes bases de formas (MPEG-7 e MNIST) são apresentados para ilustrar e validar os métodos. TSS demonstra resultados comparáveis aos métodos estado da arte, que geralmente dependem de algoritmos custosos de otimização de correspondências. Já o TSB, com sua função de distância em tempo linear, se demonstra como uma solução adequada para grandes coleções de formas. / In this work, two new shape descriptors are proposed for tasks in Content-Based Image Retrieval (CBIR) and Shape Analysis tasks, which are built upon an extended tensor scale based on the Euclidean Distance Transform (EDT). First, the tensor scale algorithm is applied to extract shape attributes from its local structures (thickness, orientation, and anisotropy) as represented by the largest ellipse within a homogeneous region centered at each image pixel. In the new descriptors, the upper limit of the interval of local orientation of tensor scale ellipses is extended from 180º to 360º, to better discriminate the description of local structures. Then, the new descriptors are built based on different sampling approaches, aiming to summarize the most relevant features. In the first proposed descriptor, Tensor Scale Sector descriptor (TSS), the local distributions of relative orientations within circular sectors are used to compose a fixed-length feature vector, for a region-based shape characterization. For the second method, the Tensor Scale Band (TSB) descriptor, histograms of relative orientations are considered for each circular concentric band, to also compose a fixed-length feature vector, with linear time distance function for matching. Experimental results for different shape datasets (MPEG-7 and MNIST) are presented to illustrate and validate the methods. TSS can achieve high retrieval values comparable to state-of-the-art methods, which usually rely on time-consuming correspondence optimization algorithms, but uses a simpler and faster distance function, while the even faster linear complexity of TSB leads to a suitable solution for very large shape collections.

Osteoporosis is a common bone disease that increases risk of low-trauma fractures associated with substantial morbidity, mortality, and financial costs. Clinically, osteoporosis is defined by low bone mineral density (BMD). BMD explains approximately 60-70% of the variance in bone strength. The remainder is due to the cumulative and synergistic effects of other factors, including trabecular and cortical bone micro-architecture. In vivo quantitative characterization of trabecular bone (TB) micro-architecture with high accuracy, reproducibility, and sensitivity to bone strength will improve our understanding of bone loss mechanisms and etiologies benefitting osteoporotic diagnostics and treatment monitoring processes. The overall aim of the Ph.D. research is to design, develop and evaluate new 3-D imaging processing algorithms to characterize the quality of TB micro-architectural in terms of topology, orientation, thickness and spacing, and to move the new technology from investigational research into the clinical arena. Two algorithms regarding to this purpose were developed and validated in detail - (1) star-line-based TB thickness and marrow spacing computation algorithm, and (2) tensor scale (t-scale) based TB topology and orientation computation algorithm. The TB thickness and marrow spacing algorithm utilizes a star-line tracing technique that effectively accounts for partial voluming effects of in vivo imaging with voxel size comparable to TB thickness and also avoids the problem of digitization associated with conventional algorithms. Accuracy of the method was examined on computer-generated phantom images while the robustness of the method was evaluated on human ankle specimens in terms of stability across a wide range of resolutions, repeat scan reproducibility under in vivo condition, and correlation between thickness values computed at ex vivo and in vivo resolutions. Also, the sensitivity of the method was examined by its ability to predict bone strength of cadaveric specimens. Finally, the method was evaluated in an in vivo human study involving forty healthy young-adult volunteers and ten athletes. The t-scale based TB topology and orientation computation algorithm provides measures characterizing individual trabeculae on the continuum between perfect plate and perfect rod as well as individual trabecular orientation. Similar to the TB thickness and marrow spacing computation algorithm, accuracy was examined on computer-generated phantoms while robustness of the algorithm across ex vivo and in vivo resolution, repeat scan reproducibility, and the sensitivity to experimental mechanical bone strength were evaluated in a cadaveric ankle study. And the application of the algorithm was evaluated in a human study involving forty healthy young-adult volunteers and ten patients with SSRI treatment. Beside these two algorithms, an image thresholding algorithm based on the class uncertainty theory is developed to segment TB structure in CT images. Although the algorithm was developed for this specific application, it also works effectively for general 2-D and 3-D images. Moreover, the class uncertainty theory can be utilized as adaptive information in more sophisticated image processing algorithms such as Snakes, ASMs and graph search.